More than that, the variance can also be used to detect quantum
entanglement [1]}, [2]}, [3]}, [4]},  and quantify quantum coherence , [5]}, [6]}. In view of the importance of the variance
in quantum information theory, we aim to explore the role played by
the variance, as a unifying tool, in characterizing
the quantumness. In a \(d\) -dimensional system with fixed
basis \(\lbrace |i\rangle :i=0,1,\ldots ,d-1\rbrace \) , any pure state can be
expressed as \(|\psi \rangle =\sum _{i=0}^{d-1} \psi _i |i\rangle \) . Since the
quantumness of the pure state is invariant under relabeling the
coefficients \(\lbrace \psi _i\rbrace \) , in order to study the quantumness of
the pure state \(|\psi \rangle \) , one may consider the averaged
quantumness in the set of pure states \(\lbrace P_{\pi }|\psi \rangle \rbrace \) ,
where \(P_{\pi }\)  are all possible permutation matrices
corresponding to the permutations \(\pi \)  of the permutation group
\(S_d\) . In this context, we propose the averaged variances of any
diagonal observable over the set of \(\lbrace P_{\pi }|\psi \rangle \rbrace \)  and
named as symmetrized variance \(\widehat{\operatorname{Var}}(|\psi \rangle , A)\) .
By analysis and calculation, we obtain the analytical expression of the symmetrized variance and find that the information of the pure state and the diagonal observable are factorized.
By shifting the information of the diagonal observable, we introduce the core notion of this paper called the standard symmetrized variance \(\widehat{V}(|\psi \rangle )\) .
